Ising model

About: Ising model is a research topic. Over the lifetime, 25508 publications have been published within this topic receiving 555000 citations.

Facilitated kinetic Ising models and the glass transition

Abstract: The class of ‘‘facilitated’’ kinetic Ising models introduced in a recent letter is investigated in greater detail An interpretation of the models in terms of relaxation processes in dense fluids is described Various types of kinetic behavior are predicted for the different spin models: (A) Arrhenius temperature dependence of the average structural relaxation time, (B) non‐Arrhenius temperature dependence with a divergence of the relaxation time at a nonzero temperature, and (C) non‐Arrhenius temperature dependence with a divergent relaxation time only at zero temperature All of the models show nonexponential decay of equilibrium time correlation functions, consistent with the Kohlrausch–Williams–Watts empirical form The nature of the glass transitions exhibited by the various models is discussed

A Theory of Cooperative Phenomena. III. Detailed Discussions of the Cluster Variation Method

Abstract: A series of approximations for the statistical mechanics of order‐disorder, proposed by Bethe, Takagi, Yang‐Li‐Hill, Kikuchi, and others, are investigated in detail in two ways. (1) A new interpretation of the method for constructing the combinatory factor is presented in order to give a better understanding of the nature of approximations. (2) The partition functions with approximate combinatory factors are expanded to compare with the rigorous expansion and the discrepancies between them are investigated in detail. One of the conclusions is that in order to obtain a higher approximation, it is necessary to use the basic figure ``closed'' with respect to the cluster of the preceding approximation. In appendices, an improved treatment of the body‐centered cubic lattice (Ising model) is given, and Bethe's fundamental assumptions are derived from our scheme.

Geometric frustration in buckled colloidal monolayers

Abstract: Geometric frustration arises when lattice structure prevents simultaneous minimization of local interaction energies. It leads to highly degenerate ground states and, subsequently, to complex phases of matter, such as water ice, spin ice, and frustrated magnetic materials. Here we report a simple geometrically frustrated system composed of closely packed colloidal spheres confined between parallel walls. Diameter-tunable microgel spheres are self-assembled into a buckled triangular lattice with either up or down displacements, analogous to an antiferromagnetic Ising model on a triangular lattice. Experiment and theory reveal single-particle dynamics governed by in-plane lattice distortions that partially relieve frustration and produce ground states with zigzagging stripes and subextensive entropy, rather than the more random configurations and extensive entropy of the antiferromagnetic Ising model. This tunable soft-matter system provides a means to directly visualize the dynamics of frustration, thermal excitations and defects.

Regularity Properties and Pathologies of Position-Space Renormalization-Group Transformations

Abstract: We reconsider the conceptual foundations of the renormalization-group (RG) formalism, and prove some rigorous theorems on the regularity properties and possible pathologies of the RG map. Regarding regularity, we show that the RG map, defined on a suitable space of interactions (= formal Hamiltonians), is always single-valued and Lipschitz continuous on its domain of definition. This rules out a recently proposed scenario for the RG description of first-order phase transitions. On the pathological side, we make rigorous some arguments of Griffiths, Pearce and Israel, and prove in several cases that the renormalized measure is not a Gibbs measure for any reasonable interaction. This means that the RG map is ill-defined, and that the conventional RG description of first-order phase transitions is not universally valid. For decimation or Kadanoff transformations applied to the Ising model in dimension $d \ge 3$, these pathologies occur in a full neighborhood $\{ \beta > \beta_0 ,\, |h| < \epsilon(\beta) \}$ of the low-temperature part of the first-order phase-transition surface. For block-averaging transformations applied to the Ising model in dimension $d \ge 2$, the pathologies occur at low temperatures for arbitrary magnetic-field strength. Pathologies may also occur in the critical region for Ising models in dimension $d \ge 4$. We discuss in detail the distinction between Gibbsian and non-Gibbsian measures, and give a rather complete catalogue of the known examples. Finally, we discuss the heuristic and numerical evidence on RG pathologies in the light of our rigorous theorems.